AP Calculus (BC) Chapter 10 Test No Calculator Section Name: Date: Period:
Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The graph in the xy-plane represented by x = 3 sin t and y = 2 cos t is (A) a circle (B) an ellipse (C) a hyperbola (D) a parabola (E) a line 2. The area enclosed by the polar equation r = 4 + cos θ, for 0 θ 2π, is (A) 0 (B) 9π 2 (C) 18π (D) 33π 2 (E) 33π 4 3. Find the length of the arc of the curve defined by x = 1 2 t2 and y = 1 9 (6t + 9) 3 2, from t = 0 to t = 2. (A) 8 (B) 10 (C) 12 (D) 14 (E) 16
4. The area of the region inside the polar curve r = 4 sin θ but outside the polar curve r = 2 2 is given by (A) 2 (B) 1 2 (C) 1 2 (D) 1 2 (E) 1 2 3π/4 π/4 3π/4 π/4 3π/4 π/4 3π/4 π/4 3π/4 π/4 (4 sin 2 θ 1) dθ (4 sin θ 2 2) 2 dθ (4 sin θ 2 2) dθ (16 sin 2 θ 8) dθ (4 sin 2 θ 1) dθ. 5. If, for t > 0, x = t 2 and y = cos t 2, then dy dx = (A) cos t 2 (B) sin t 2 (C) sin 2t (D) sin t 2 (E) cos 2t
Part II. Free-Response Questions 1. A particle moves in the xy-plane so that its position at any time t, for π t π, is given by x(t) = sin 3t and y(t) = 2t. y (a) (4 points) Sketch the path of the particle in the xy-plane provided. Indicate the direction of motion along the path. 1 x (b) (4 points) Find the range of x(t) and the range of y(t). (c) (4 points) Find the smallest positive value of t for which the x-coordinate of the particle is a local maximum. What is the speed of the particle at this time? (d) (4 points) Write down an integral which will compute the distance traveled by the particle from t = π to π.
put polar eqn here 2. The figure above shows the graphs of the line x = 5 y and the curve C is given by 3 x = 1 + y 2. Let S be the region bounded by the two graphs and the x-axis. The line and the curve intersect at point P. (a) (5 points) Curve C is a part of the curve x 2 y 2 = 1. Show that x 2 y 2 = 1 can be written as the polar equation r 2 1 = cos 2 θ sin 2 θ. (b) (4 points) Use the polar equation given above to set up an integral expression with respect to the polar angle θ that represents the area of S.
AP Calculus (BC) Chapter 10 Test Calculators Allowed Name: Date: Period:
Part III. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The length of the curve determined by x = 2t 3 and y = t 3 from t = 0 to t = 1 is (A) 5 7 5 (B) 2 (C) 3 2 (D) 5 (E) 3 2. A particle moves along a path described by x = cos 3 t and y = sin 3 t. The distance that the particle travels along the path from t = 0 to t = π 2 is (A) 0.75 (B) 1.50 (C) 0 (D) 3.50 (E) 0.75 3. Find the area inside one loop of the curve r = sin 2θ. (A) (B) (C) π 4 π 16 π 8 (D) π 2 (E) π
4. A particle moves on a plane curve so that at any time t > 0 its position is defined by the parametric equations x(t) = 3t 2 7 and y(t) = 4t2 + 1. The acceleration vector 3t of the particle at t = 2 is ( (A) 6, 1 ) 12 ( (B) 17, 17 ) 6 ( (C) 12, 47 ) 12 ( (D) 12, 33 ) 12 ( (E) 6, 17 ) 6 5. The acceleration of a particle is described by the parametric equation x (t) = t2 4 + t and y (t) = 1. If the velocity vector of the particle when t = 2 is (4, ln 2), what is the 3t velocity vector of the particle when t = 1? ( 5 (A) 4, 1 ) 3 ( 23 (B) 12, ln 4 ) 3 ( 23 (C) 12, ln 2 ) 3 ( 5 (D) 4, 2 ) 3 ln 2 ( 23 (E) 12, 4 ) 3 ln 2
Part IV. Free-Response Questions 1. The figure to the right shows the path traveled by a roller coaster car over the time interval 0 t 18 seconds. The position and velocity of the car at time t can be modeled parametrically by put rollercoaster here Position Velocity x(t) = 10t + 4 sin t x (t) = 10 + 4 cos t y(t) = (20 t)(1 cos t) y (t) = (20 t) sin t + cos t 1 where x and y are measured in meters and t is measured in seconds. (a) (4 points) Find the slope of the path at time t = 2. Show the computations that lead to your answer. (b) (4 points) Find the time t at which the car is at its maximum height, and find the speed, in m/sec, of the car at this time. (c) (4 points) For 0 < t < 18, there are two times at which at the car is at ground level (y = 0). Find these two times and write an expression that gives the average speed, in m/sec, of the car between these two times.
2. The curve above is drawn in the xy-plane and is described by the equation in polar coordinates r = θ + sin(2θ), 0 θ π, where r is measured in meters and θ is measured in radians. The derivative of r with respect to θ is given by dr dθ = 1 + 2 cos(2θ). put polargraph here (a) (5 points) Find the area of the region below the curve, to the left of the y-axis, and above the x-axis. (b) (4 points) For π 3 < θ < 2π 3, dr is negative. What does this fact say about r? dθ What does this fact say about the curve? (c) (4 points) Find the value of θ in the interval 0 θ π that corresponds to the 2 point on the curve in the first quadrant with greatest distance from the origin. Justify your answer.